Kerry Back
BUSI 721, Fall 2022
JGSB, Rice University
Tesla went down 50% between Nov 2021 and May 2022.
It then went up 50% between May 2022 and Aug 2022.
For each $100 of Tesla stock, shareholders experienced
100 → 50 → 75
They lost 25%, even though the arithmetic average return was zero.
So, lose 50% and make 50% → lose 25%. Suppose you
- make 50% and then lose 50%?
- lose 50% and then make 100%?
- make 100% and then lose 50%?
Given returns \(r_1, \cdots, r_n,\)
the geometric average return is the number r such that
\((1+r)^{n}=(1+r_1)\cdots(1+r_{n})\)
So earning r each period produces the same accumulation as the actual returns \(r_1, \cdots, r_n.\)
We solve for it as
\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]
Examples
make 50% and lose 50% → geometric average is \[\sqrt{1.5 x 0.5}-1=-0.134\]
make 100% and lose 50% → geometric average is \[\sqrt{2 x 0.5}-1=0\]